If the mental can affect, or be affected by, the physical, then the mental must itself be physical. Otherwise the physical world would not be explanatorily closed. But it is closed. There are reasons to hold that materialism (in both its reductive and non-reductive varieties) is false. So how are we to explain the apparent responsiveness of the physical to the mental and vice versa? The only possible solution seems to be this: physical objects are really projections or isomorphs of objects whose essential properties are mental. (A slightly less accurate way of putting this would be to say: the constitutive - i.e. the non-structural and non-phenomenal - properties of physical objects are mental, i.e. are such as we are used to encountering only in "introspection".) The chair, qua thing that I can know through sense perception, and through hypotheses based strictly thereupon, is a kind of shadow of an object that is exactly like it, except that this other objects essential properties are mental. This line of thought, though radically counterintuitive, explains the apparent responsiveness of the mental to the physical, and vice versa, without being open to any of the criticisms to which materialism, dualistic interaction ism, and epiphenomenalism are open.
Can Neuroscientists Test a New Physicalist Mind/Body View: DiCoToP (Diachronic Conjunctive Token Physicalism)?